Q. 50 - Calculus

Question

Use definite integrals to find the volume of each solid of revolution described in Exercises 49–61. (It is your choice whether to use disks/washers or shells in these exercises.)  

The region between the graph of $f\left(x\right)=e^{2x}$ and the x-axis on [0, 4], revolved around the x-axis. 

Step-by-Step Solution

Answer

The integral does not exist

Step 1: Given information

We are given a function $e^{2x}$

Step 2: FInd the integral and evaluate it

We know that the integral can be given as 

$V=2\pi {\int }_c^{d}r\left(y\right)h\left(y\right)dy$. The axis of revolution is x-axis hence the radius is $r\left(y\right)=y$ and the height can be given as $h\left(y\right)=\frac{1}{2}\ln y$ and substituting the value in the integral we get,

$$\begin{gathered} V=\pi {\int }_0^{4}y\left(\ln y\right)dy \\ V=\pi {{[\frac{y^2}{2}\ln y-\frac{y^2}{4}]}^4}_{0}integral does not exist \end{gathered}$$